Proto languages are usually unwritten but can be reconstructed to a certain point based on what they left behind in “daughter languages”. But, I feel that Latin (classical and vulgar) are part of one proto language: proto romance. Is this a valid theory? Explain.
"Proto-X" tends to be used for the last common ancestor of X—the point at which the X languages started to diverge and become their own entities. This is why you'll sometimes hear about "pre-Proto-Indo-European": PIE is the last common ancestor of the Indo-European languages, and "pre-PIE" is what might have come before it. (It mostly comes up in discussions of "this feature of PIE is very weird, it makes more sense if it used to be like this in pre-PIE, and then evolved into that".)
So, Proto-Romance is the last common ancestor of the Romance languages (*): the particular variety of Vulgar Latin that was spoken at a particular point in the Roman Empire, before all the dialects diverged enough to be considered different languages. Classical Latin and Vulgar Latin itself had diverged from a common ancestor, which I suppose you could call "Proto-Latin", a few centuries before that. And before that was Proto-Latino-Faliscan, and then Proto-Italic, and then Proto-Italo-Venetic, and then (arguably) Proto-Italo-Celtic, and so on.
(*) Except Sardinian/Sard, which may or may not be considered a Romance language—it did come from Latin, but diverged at a very different point than any of the others, which one might call "Proto-Sardinio-Romance" or something like that.